US 8,510,084 B2Granted

Method, apparatus and computer program for multiple time stepping simulation of a thermodynamic system using shadow hamiltonians

AssigneeFujitsu Limited

InventorsElena Vitalievna Akhmatskaya, Sebastian Reich

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Abstract

A computer-implemented method including a momentum refreshment process and a conservative dynamics process, wherein the refreshment process including carrying out an operation of mixing the collective momentum vector with a noise vector and carrying out an acceptance/rejection (acceptance and rejection) operation; the dynamics process including applying a multiple time stepping conservative dynamics operation to a current state, in which calculations corresponding to more slowly varying energy parts undergo an averaging procedure and are carried out at a larger time step than calculations corresponding to more quickly varying energy parts; and carrying out an acceptance/rejection operation based on an approximation of the system energy expressed using shadow Hamiltonians and include accepting a current state or returning a replacement state.

Claims14

Dependent claims

A computer-implemented method of simulating behaviour of a thermodynamic system over time, the thermodynamic system having potential energy that can be split into more quickly varying parts and more slowly varying parts and having a state described by collective vectors of position and momentum at any given time, the method comprising:

a momentum refreshment process and a conservative dynamics process performed by a processor, wherein
the momentum refreshment process comprises carrying out an operation of mixing the collective momentum vector with a noise vector and carrying out an acceptance and rejection operation;
the conservative dynamics process comprises applying a multiple time stepping conservative dynamics operation to a current state, in which operation calculations for forces corresponding to more slowly varying energy parts in the thermodynamic system undergo an averaging procedure and are carried out at a larger time step than calculations for forces corresponding to more quickly varying energy parts;
and carrying out an acceptance and rejection operation;
wherein
the acceptance and rejection operations are based on an approximation of the system energy expressed using shadow Hamiltonians and comprise accepting a current state or returning a replacement state,
wherein the states in the method are denoted by Ωi=(YiT,ti)T, i=0, . . . , I, where I is a given integer, Yi=(XiT,1,PiT,bi)T, Xi is a collective vector of atomic positions, Pi is a collective vector of atomic momenta, bi is a scalar, and ti is time and wherein the mixing operation comprises:
given a current state, mixing its collective atomic momentum vector Pi with an independent and identically distributed normal noise vector Ξi of dimension 3N, so that
{tilde over (P)}i=cos(φ)Pi+sin(φ)Ξi,
{tilde over (Ξ)}i=cos(φ)Ξi−sin(φ)Pi,
where i is a given integer, N is the number of particles in the system, 0<φ≦π/2 is a given angle, Ξi˜N[0,βM−1], N[0,βM−1] denotes the (3N)-dimensional normal distribution with zero mean and covariance matrix βM−1, M is the diagonal mass matrix of the molecular system, and β=1/kBT is the inverse temperature, proposed vector being denoted by {tilde over (Ω)}i=({tilde over (Y)}iT,ti)T, {tilde over (Y)}i=(XiT,1,{tilde over (P)}iT,bi)T;
and preferably wherein the subsequent acceptance/rejection operation comprises obtaining the resulting state Ωi;
through a Metropolis accept/reject criterion:
⁢
Ω
_
i
=
{
Ω
~
i
with
⁢
⁢
probability
⁢
⁢
min
(
1
,
exp
⁡
(
-
βΔ
⁢
⁢
H
Δ
⁢
⁢
t
e
)
Ω
i
otherwise
,
⁢
⁢
⁢
with
Δ
⁢
⁢
H
Δ
⁢
⁢
t
e
:=
H
Δ
⁢
⁢
t
⁡
(
X
i
,
P
~
i
)
+
1
2
⁢
Ξ
~
i
T
⁢
M
-
1
⁢
Ξ
~
i
-
H
Δ
⁢
⁢
t
⁡
(
X
i
,
P
i
)
-
1
2
⁢
Ξ
i
T
⁢
M
-
1
⁢
Ξ
i
,
in which HΔt is the shadow Hamiltonian and ΔHΔte is the change in shadow Hamiltonian due to the mixing operation.

121 more elements

An apparatus which simulates behaviour of a thermodynamic system over time, the thermodynamic system having potential energy that can be split into more quickly varying parts and more slowly varying parts and having a state described by collective vectors of position and momentum at any given time;

the apparatus comprising:
a processor, comprising:
a momentum refreshment part operable to carry out an operation of mixing the collective momentum vector with a noise vector and an acceptance and rejection operation;
a conservative dynamics part operable to apply a multiple time stepping conservative dynamics operation to a current state, in which operation calculations for forces corresponding to more slowly varying energy parts in the thermodynamic system undergo an averaging procedure and are carried out at a larger time step than calculations for forces corresponding to more quickly varying energy parts;
and to carry out an acceptance and rejection operation;
wherein
the acceptance and rejection operations are based on an approximation of the system energy expressed using shadow Hamiltonians and comprise accepting a current state or returning a replacement state,
wherein the states are denoted by Ωi=(YiT,ti)T, i=0, . . . , I, where I is a given integer, Yi=(XiT,1,PiT,bi)T, Xi is a collective vector of atomic positions, Pi is a collective vector of atomic momenta, bi is a scalar, and ti is time and wherein the mixing operation comprises:
given a current state, mixing its collective atomic momentum vector Pi with an independent and identically distributed normal noise vector Ξi of dimension 3N, so that
{tilde over (P)}i=cos(φ)Pi+sin(φ)Ξi,
{tilde over (Ξ)}i=cos(φ)Ξi−sin(φ)Pi,
where i is a given integer, N is the number of particles in the system, 0<φ≦π/2 is a given angle, Ξi˜N[0,βM−1], N[0,βM−1] denotes the (3N)-dimensional normal distribution with zero mean and covariance matrix βM−1, M is the diagonal mass matrix of the molecular system, and β=1/kBT is the inverse temperature, proposed vector being denoted by {tilde over (Ω)}i=({tilde over (Y)}iT,ti)T, {tilde over (Y)}i=(XiT,1,{tilde over (P)}iT,bi)T;
and preferably wherein the subsequent acceptance/rejection operation comprises obtaining the resulting state Ωi;
through a Metropolis accept/reject criterion:
Ω
_
i
=
{
Ω
~
i
with
⁢
⁢
probability
⁢
⁢
min
⁢
⁢
(
1
,
exp
⁡
(
-
βΔ
⁢
⁢
H
Δ
⁢
⁢
t
e
)
Ω
i
otherwise
,
⁢
⁢
with
⁢
⁢
Δ
⁢
⁢
H
Δ
⁢
⁢
t
e
:=
H
Δ
⁢
⁢
t
⁡
(
X
i
,
P
~
i
)
+
1
2
⁢
Ξ
~
i
T
⁢
M
-
1
⁢
Ξ
~
i
-
H
Δ
⁢
⁢
t
⁡
(
X
i
,
P
i
)
-
1
2
⁢
Ξ
i
T
⁢
M
-
1
⁢
Ξ
i
,
in which HΔt is the shadow Hamiltonian and ΔHΔte is the change in shadow Hamiltonian due to the mixing operation.

124 more elements

A non-transitory computer-readable medium storing a computer program which, when executed on a processor, carries out a computer-implemented method of simulating behaviour of a thermodynamic system over time, the thermodynamic system having potential energy that can be split into more quickly varying parts and more slowly varying parts and having a state described by collective vectors of position and momentum at any given time, the method comprising a momentum refreshment process and a conservative dynamics process, wherein

the momentum refreshment process comprises carrying out an operation of mixing the collective momentum vector with a noise vector and carrying out an acceptance and rejection operation;
the conservative dynamics process comprises applying a multiple time stepping conservative dynamics operation to a current state, in which operation calculations for forces corresponding to more slowly varying energy parts in the thermodynamic system undergo an averaging procedure and are carried out at a larger time step than calculations for forces corresponding to more quickly varying energy parts;
and carrying out an acceptance and rejection operation;
wherein
the acceptance and rejection operations are based on an approximation of the system energy expressed using shadow Hamiltonians and comprise accepting a current state or returning a replacement state,
wherein the states are denoted by Ωi=(YiT,ti)T, i=0, . . . , I, where I is a given integer, Yi=(XiT,1,PiT,bi)T,Xi is a collective vector of atomic positions, Pi is a collective vector of atomic momenta, bi is a scalar, and ti is time and wherein the mixing operation comprises:
given a current state, mixing its collective atomic momentum vector Pi with an independent and identically distributed normal noise vector Ξi of dimension 3N, so that
{tilde over (P)}i=cos(φ)Pi+sin(φ)Ξi,
{tilde over (Ξ)}i=cos(φ)Ξi−sin(φ)Pi,
where i is a given integer, N is the number of particles in the system, 0<φ≦π/2 is a given angle, Ξi˜N[0,βM−1], N[0,βM−1] denotes the (3N)-dimensional normal distribution with zero mean and covariance matrix βM−1, M is the diagonal mass matrix of the molecular system, and β=1/kBT is the inverse temperature, proposed vector being denoted by {tilde over (Ω)}i=({tilde over (Y)}iT,ti)T, {tilde over (Y)}i=(XiT,1,{tilde over (P)}iT,bi)T;
and preferably wherein the subsequent acceptance/rejection operation comprises obtaining the resulting state Ωi;
through a Metropolis accept/reject criterion:
Ω
_
i
=
{
Ω
~
i
with
⁢
⁢
probability
⁢
⁢
min
⁢
⁢
(
1
,
exp
⁡
(
-
βΔ
⁢
⁢
H
Δ
⁢
⁢
t
e
)
Ω
i
otherwise
,
⁢
⁢
with
⁢
⁢
Δ
⁢
⁢
H
Δ
⁢
⁢
t
e
:=
H
Δ
⁢
⁢
t
⁡
(
X
i
,
P
~
i
)
+
1
2
⁢
Ξ
~
i
T
⁢
M
-
1
⁢
Ξ
~
i
-
H
Δ
⁢
⁢
t
⁡
(
X
i
,
P
i
)
-
1
2
⁢
Ξ
i
T
⁢
M
-
1
⁢
Ξ
i
,
in which HΔt is the shadow Hamiltonian and ΔHΔte is the change in shadow Hamiltonian due to the mixing operation.

122 more elements

Dependencies are inferred from the claim text.